Degenerate degree distribution for infinite generalized Malthusian parameter

Let $\mathcal{T}$ be a $(\mu,f,\ell)$-RIF tree, and define

$$\alpha=\inf\left\{\lambda>0:\ell\,\mathbb{E}\left[\sum_{n=1}^{\infty}\prod_{i=0}^{n-1}\frac{f(i,W)}{f(i,W)+\lambda}\right]<\infty\right\}.$$

Assume that $\alpha=\infty$, and let $N_0(t,B)$ denote the number of degree-zero vertices present by time $t$ whose fitness lies in the measurable set $B$. Infinite-parameter degree distribution conjecture. For every measurable set $B$,

$$\frac{N_0(t,B)}{\ell t}\xrightarrow[t\to\infty]{}\mu(B)$$

almost surely. This is the degenerate limiting case of the proposed universal degree distribution formula, asserting that when the generalized Malthusian parameter is infinite, asymptotically all vertices have degree zero with fitness distribution $\mu$.